Problem: Determine how many solutions exist for the system of equations. ${3x+y = 5}$ ${9x+3y = 15}$
Solution: Convert both equations to slope-intercept form: ${3x+y = 5}$ $3x{-3x} + y = 5{-3x}$ $y = 5-3x$ ${y = -3x+5}$ ${9x+3y = 15}$ $9x{-9x} + 3y = 15{-9x}$ $3y = 15-9x$ $y = 5-3x$ ${y = -3x+5}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -3x+5}$ ${y = -3x+5}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${3x+y = 5}$ is also a solution of ${9x+3y = 15}$, there are infinitely many solutions.